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vmbo havo vwo

Parabola

After this simulation you can explain what a, b and c do to the graph of y = ax² + bx + c.

Level
Mode
Parabola y = x² − 2x − 3 Graph with its vertex at (1; −4), crossing the x-axis at x = −1 and x = 3.
y= ax2 +bx +c
a
b
c

Start: a = 1, b = −2, c = −3 → D = 16, roots x = −1 and x = 3, vertex (1; −4).

What do you see? a = 1,0, b = −2,0, c = −3,0. Discriminant D = 16,0. Two roots: x = −1,0 and 3,0. Vertex at (1,0, −4,0). The parabola opens upwards.
Table x / y
Values on the parabola
xy
−6,045,0
−4,526,3
−3,012,0
−1,52,3
0,0−3,0
1,5−3,8
3,00,0
4,58,3
6,021,0
Settings

Explanation

A parabola is the graph of a quadratic function y = ax² + bx + c with a ≠ 0. The sign of a sets the direction: a positive a opens upwards, a negative a opens downwards, and a larger |a| makes the curve narrower. c is where the graph crosses the y-axis. b moves the vertex sideways, because the vertex lies at x = −b/(2a).

In Dutch schools, vmbo mostly uses y = ax² + c, without the bx term. At havo and vwo you work with the full form and find the roots (nulpunten) through the discriminant. The formulas are the same ones the quadratic formula calculator uses.

Worked example

The worked example uses y = x² − 2x − 3, so a = 1, b = −2 and c = −3.

  1. DiscriminantD = (−2)² − 4·1·(−3) = 4 + 12 = 16
  2. Rootsx = (2 ± 4) / 2 → x = 3 or x = −1
  3. Vertexx = −(−2)/(2·1) = 1, y = 1 − 2 − 3 = −4 → vertex (1; −4)

Compare with the graph above: the curve crosses the x-axis at −1 and 3, and its lowest point is (1; −4).

Tasks

Pick the Tasks mode above the graph, or work through them on paper:

  1. Make D less than 0, so the parabola does not cross the x-axis.
  2. Make a negative so the parabola opens downwards.
  3. Put the vertex on the y-axis (b = 0).
  4. Make the parabola touch the x-axis (D = 0).
  5. Set a = 1 and move the vertex to x = 2. Choose b yourself.

In Predict mode you choose an outcome before the change is applied. In Challenge mode you match a dashed target curve.

Common mistakes

Wrong

Keeping a = 0 and still calling the graph a parabola.

Right

a ≠ 0. With a = 0 you are left with a straight line; see the linear equation solver.

Wrong

Reading D < 0 as “I made a calculation error”.

Right

D < 0 means the graph has no intersection with the x-axis among the real numbers.

Wrong

Flipping the sign of b in the vertex formula.

Right

The vertex x is −b/(2a). With b = −2, −b is positive.

Formula sheet

Print-friendly: use Print in your browser.

  • Parabola y=ax2+bx+c
  • Discriminant D=b2−4ac
  • Quadratic formula x=−b±D2a
  • Vertex x x=−b2a
  • Check D for y = x² + x + 1 → D = −3 D=1−4=−3
Discriminant

D = b² − 4ac. With D < 0 there are no real roots.

abc-formule

The Dutch name for the quadratic formula for ax² + bx + c = 0.

What you need to know

Linked to the 2027 central exam syllabus. Not a full exam list.

havo · wiskunde B · domain B (B2)

Functies, grafieken en vergelijkingen · syllabus wiskunde B havo 2027 (Examenblad)

vwo · wiskunde B · domain B (B5)

Functies, grafieken en vergelijkingen · syllabus wiskunde B vwo 2027 (Examenblad)

Test yourself

Practice with the quadratic formula practice module.

Work it out

Need the roots and steps for ax² + bx + c = 0? Use the quadratic formula calculator.

Practice with real exam questions on Examenblad.nl (in Dutch). We do not host past exams.

Questions

What does a do to a parabola?

a decides whether the parabola opens upwards (a > 0) or downwards (a < 0), and how narrow or wide the curve is.

Where is the vertex?

The vertex is at x = −b/(2a). Put that x into y = ax² + bx + c to get the y-value.

What if D is less than 0?

Then the graph does not cross the x-axis: there are no real roots. Example: y = x² + x + 1 has D = −3.

What happens when a = 0?

Then a straight line y = bx + c is left. Use the linear equation solver instead.